# Compute Elasticity Tensor

Compute an elasticity tensor.

## Description

The material ComputeElasticityTensor builds the elasticity (stiffness) tensor with various user-selected material symmetry options. ComputeElasticityTensor also rotates the elasticity tensor during the initial time step only; this class does not rotate the elasticity tensor during the simulation. The initial rotation is only performed if the user provides arguments to the three Euler angle parameters; the Bunge Euler angles provided in this class are used to perform passive (from the sample to the crystal) rotations.

For a general stiffness tensor with 21 independent components, the elasticity tensor within the tensor mechanics module can be represented with the notation shown in Eq. 1. Nonetheless, the full Rank-4 tensor with all 81 components is created by ComputeElasticityTensor. (1)

There are several different material symmetry options that a user can apply to build the elasticity tensor for a mechanics simulation that are discussed below.

## General Symmetry

The fill method symmetric21 is used to create the elasticity tensor for a linear hyperelastic material with 21 independent components: the symmetries shown in Eq. 2 are used to determine the independent components (Slaughter, 2012). (2)

### Example Input File Syntax

[./elasticity_tensor]
type = ComputeElasticityTensor
fill_method = symmetric21
C_ijkl = '1111 1122 1133 1123 1113 1112 2222 2233 2223 2213 2212 3333 3323 3313 3312 2323 2313 2312 1313 1312 1212'
[../]
(moose/modules/combined/test/tests/linear_elasticity/tensor.i)

which shows the expected order of the elasticity tensor components in the input argument string.

## Orthotropic Symmetry

The fill method symmetric9 is appropriate for materials with three orthotropic planes of symmetry (Malvern, 1969), and is often used for simulations of anistropic materials such as cubic crystals. The enginering elasticity tensor notation, Eq. 1, for an orthotropic material is given in Eq. 3 (3)

### Example Input File Syntax

[./elasticity_tensor]
type = ComputeElasticityTensor
C_ijkl = '1.684e5 0.176e5 0.176e5 1.684e5 0.176e5 1.684e5 0.754e5 0.754e5 0.754e5'
fill_method = symmetric9
[../]
(moose/modules/tensor_mechanics/test/tests/finite_strain_elastic/finite_strain_elastic_new_test.i)

In the Einstein index notation shown in Eq. 1, the parameter C_ijkl expects the elasticity components in the order C_ijkl = '1111 1122 1133 2222 2233 3333 2323 3131 1212' for the symmetric9 fill method option.

## Linear Isotropic Symmetry

The two constant istropic symmetry fill methods symmetric_isotropic and symmetric_isotropic_E_nu are used in the dedicated isotropic elasticity tensor ComputeIsotropicElasticityTensor. These two fill methods use the symmetries shown in Eq. 4 to build the elasticity tensor. (4) Please see the documentation page for ComputeIsotropicElasticityTensor for details and examples of the input file syntax for linear elastic isotropic elasticity tensors.

## Antisymmetric Isotropic Symmetry

The fill method antisymmetric_isotropic is used for an antisymmetric isotropic material in a shear case. The elasticity tensor is built using the symmetries shown in Eq. 5 (5) where is the permutation tensor and is the summation index.

## Transverse Isotropic (Axisymmetric)

The fill method axisymmetric_rz is used for materials which are isotropic with respect to an axis of symmetry, such as a material composed of fibers which are parallel to the axis of symmetry (Slaughter, 2012). The engineering notation matrix in this case is shown by Eq. 6. (6)

### Example Input File Syntax

[./elasticity_tensor]
#Material constants selected to match isotropic lambda and shear modulus case
type = ComputeElasticityTensor
C_ijkl = '1022726 113636 113636 1022726 454545'
fill_method = axisymmetric_rz
[../]
(moose/modules/tensor_mechanics/test/tests/isotropic_elasticity_tensor/2D-axisymmetric_rz_test.i)

In the Einstein index notation shown in Eq. 1, the parameter C_ijkl expects the elasticity components in the order C_ijkl = '1111, 1122, 1133, 3333, 2323' for the axisymmetric_rz fill method option.

## Principal Directions for Stress and Strain

The fill method principal is appropriate for the case when the principal directions of strain and stress align. The engineering notation representation of the elasticity tensor is shown in Eq. 7. (7)

In the Einstein index notation shown in Eq. 1, the parameter C_ijkl expects the elasticity components in the order C_ijkl = '1111 1122 1133 2211 2222 2233 3311 3322 3333' for the principal fill method option.

## Cosserat Elasticity Specific Fill Methods

The following fill methods are available within ComputeElasticityTensor, but the use cases for these methods fall within the Cosserat applications which do not preserve the equilibruim of angular momentum.

### General Isotropic Symmetry

The fill method general_isotropic is used for the case of three independent components of an elasticity tensor, Eq. 8. (8)

This fill method case is used in the child class ComputeCosseratElasticityTensor; please see the documentation for ComputeCosseratElasticityTensor for details and examples of the input file syntax.

### General Antisymmetric

The fill method antisymmetric builds an antisymmetric elasticity tensor for a shear-only case. The symmetries shown in Eq. 9 are used to create the complete tensor (9) and the engineering notation representation of the anitsymmetric elasticity tensor is given in Eq. 10. (10)

This fill method case is used in the child class ComputeCosseratElasticityTensor; please see the documentation for ComputeCosseratElasticityTensor for details and examples of the input file syntax.

### No Symmetry

The general fill method for the Compute Elasticity Tensor class does not make any assumptions about symmetry for the elasticity tensor and requires all 81 components of the stiffness tensor as an input string. This fill method case is used in the child class ComputeCosseratElasticityTensor; please see the documentation for ComputeCosseratElasticityTensor for details and examples of the input file syntax.

## Input Parameters

• C_ijklStiffness tensor for material

C++ Type:std::vector

Description:Stiffness tensor for material

### Required Parameters

• computeTrueWhen false, MOOSE will not call compute methods on this material. The user must call computeProperties() after retrieving the Material via MaterialPropertyInterface::getMaterial(). Non-computed Materials are not sorted for dependencies.

Default:True

C++ Type:bool

Description:When false, MOOSE will not call compute methods on this material. The user must call computeProperties() after retrieving the Material via MaterialPropertyInterface::getMaterial(). Non-computed Materials are not sorted for dependencies.

• elasticity_tensor_prefactorOptional function to use as a scalar prefactor on the elasticity tensor.

C++ Type:FunctionName

Description:Optional function to use as a scalar prefactor on the elasticity tensor.

• base_nameOptional parameter that allows the user to define multiple mechanics material systems on the same block, i.e. for multiple phases

C++ Type:std::string

Description:Optional parameter that allows the user to define multiple mechanics material systems on the same block, i.e. for multiple phases

• euler_angle_30Euler angle in direction 3

Default:0

C++ Type:double

Description:Euler angle in direction 3

• euler_angle_20Euler angle in direction 2

Default:0

C++ Type:double

Description:Euler angle in direction 2

• euler_angle_10Euler angle in direction 1

Default:0

C++ Type:double

Description:Euler angle in direction 1

• fill_methodsymmetric9The fill method

Default:symmetric9

C++ Type:MooseEnum

Description:The fill method

• boundaryThe list of boundary IDs from the mesh where this boundary condition applies

C++ Type:std::vector

Description:The list of boundary IDs from the mesh where this boundary condition applies

• blockThe list of block ids (SubdomainID) that this object will be applied

C++ Type:std::vector

Description:The list of block ids (SubdomainID) that this object will be applied

### Optional Parameters

• enableTrueSet the enabled status of the MooseObject.

Default:True

C++ Type:bool

Description:Set the enabled status of the MooseObject.

• use_displaced_meshFalseWhether or not this object should use the displaced mesh for computation. Note that in the case this is true but no displacements are provided in the Mesh block the undisplaced mesh will still be used.

Default:False

C++ Type:bool

Description:Whether or not this object should use the displaced mesh for computation. Note that in the case this is true but no displacements are provided in the Mesh block the undisplaced mesh will still be used.

• control_tagsAdds user-defined labels for accessing object parameters via control logic.

C++ Type:std::vector

Description:Adds user-defined labels for accessing object parameters via control logic.

• seed0The seed for the master random number generator

Default:0

C++ Type:unsigned int

Description:The seed for the master random number generator

• implicitTrueDetermines whether this object is calculated using an implicit or explicit form

Default:True

C++ Type:bool

Description:Determines whether this object is calculated using an implicit or explicit form

• constant_onNONEWhen ELEMENT, MOOSE will only call computeQpProperties() for the 0th quadrature point, and then copy that value to the other qps.When SUBDOMAIN, MOOSE will only call computeSubdomainProperties() for the 0th quadrature point, and then copy that value to the other qps. Evaluations on element qps will be skipped

Default:NONE

C++ Type:MooseEnum

Description:When ELEMENT, MOOSE will only call computeQpProperties() for the 0th quadrature point, and then copy that value to the other qps.When SUBDOMAIN, MOOSE will only call computeSubdomainProperties() for the 0th quadrature point, and then copy that value to the other qps. Evaluations on element qps will be skipped

• output_propertiesList of material properties, from this material, to output (outputs must also be defined to an output type)

C++ Type:std::vector

Description:List of material properties, from this material, to output (outputs must also be defined to an output type)

• outputsnone Vector of output names were you would like to restrict the output of variables(s) associated with this object

Default:none

C++ Type:std::vector

Description:Vector of output names were you would like to restrict the output of variables(s) associated with this object

## References

1. Lawrence E Malvern. Introduction to the Mechanics of a Continuous Medium. Prentice-Hall, 1969.[BibTeX]
2. William S Slaughter. The Linearized Theory of Elasticity. Springer Science & Business Media, 2012.[BibTeX]